Discussion on Floquet Theory: Mathieu, Hill and Related Equations

Why are we meeting?

To discuss Floquet Theory and understand Mathieu, Hill and related equations.
Prof. Jayanth, Prof. Baladitya, and I are interested in this theory.
Punyabrahma from Prof. Jayanth's laboratory has demonstrated the trapping of a magnetic sphere for use in AFMs.
The governing equation for the Magnetic Trap AFM is more complicated than the usual Mathieu/Hill equations that we have seen so far.
Prof. Baladitya is interested in wave propagation in periodic structures.
I am interested in ion traps.
All the topics mentioned above require an understanding of differential equations with periodic coefficients. The Mathieu and the Hill equations are examples of such equations. But we now have more complicated equations.
So we need to have a fundamental understanding of the subject.

Mathieu Equation:

z'' + [a - 2q cos(2ξ)] z = 0,

where prime(') indicates differentiation with respect to ξ.

Demonstrations for the Mathieu Equation:

Equation Governing the Magnetic Trap AFM of GRJ and PP:

z'' + 2ηz' + [a - 2q cos(2ξ)] z / [1 - κ cos(2ξ)] = 0,

where prime(') indicates differentiation with respect to ξ.

New parameters:

η (eta): Normalized damping constant
κ (kappa): Normalized cantilever stiffness constant
Damping parameters like η are mentioned elsewhere in literature, but ...
... κ is an absolutely new contribution from Prof. Jayanth's laboratory.

Various stability charts for this equation:

Eigenvalue locus for this equation: Animated Eigenvalue Locus

Conclusion

In order to understand what we see, we need to study Floquet's theory first.

Floquet's Theory: Study of Linear Homogeneous ODEs with Periodic Coefficients

Today, I will begin a discussion on this topic.

Example Matlab Code

Objectives:

Understand how numerical solutions are obtained using the Runge-Kutta methods. See the example code in the checkRK subfolder.
Understand how the numerically determined monodromy matrix can provide information about the solutions of Mathieu, Hill, and similar equations. See the example code in the mathieu subfolder.

exampleCode
├── checkRK
│   ├── fnMSD.m
│   ├── fnMS.m
│   ├── msdRK1.m
│   ├── msdRK2.m
│   ├── msdRK4.m
│   ├── msRK1.m
│   ├── msRK2.m
│   ├── msRK4.m
│   ├── stepRK1.m
│   ├── stepRK2.m
│   └── stepRK4.m
└── mathieu
    ├── checkStability.m
    ├── findSlowOscPeriod.m
    ├── monodromyM.m
    ├── showMathieuSolutions.m
    ├── slopeMathieu.m
    ├── solveMathieu.m
    └── stepRK4.m

Checking the Runge-Kutta Methods

Try the following in the checkRK subfolder.

msRK1(0.1, 0.2, 1, 0, 10, 400)

msRK2(0.1, 0.2, 1, 0, 10, 200)

msRK4(0.1, 0.2, 1, 0, 10, 100)

Which of the three numerical solutions is the most accurate?

Applying Floquet Theory

Try the following in the mathieu subfolder.

showMathieuSolutions(0, 0.1, 100, 1000);

showMathieuSolutions(0, 0.5, 100, 1000);

showMathieuSolutions(0, 0.907, 200, 2000);

showMathieuSolutions(0.47, 0.5, 250, 2500);

showMathieuSolutions(0.471, 0.5, 250, 2500);

showMathieuSolutions(-0.121, 0.5, 250, 2500);

showMathieuSolutions(-0.122, 0.5, 250, 2500);


checkStability(0, 0.1)
checkStability(0, 0.5)
checkStability(0, 0.907)
checkStability(0, 0.91)
checkStability(0.47, 0.5)
checkStability(0.471, 0.5)
checkStability(-0.121, 0.5)
checkStability(-0.122, 0.5)


showMathieuSolutions(0, 0.1, 100, 1000);
% 88.76
findSlowOscPeriod(0, 0.1)
% 88.683

showMathieuSolutions(0, 0.1, 100, 1000);
% 16.6
findSlowOscPeriod(0, 0.1)
% 16.81

Dehmelt's Approximation

Even though we have discussed numerical techniques for calculating the solution of the Mathieu equation and other quantities associated with it, for small values of a and q, there is an approximation due to Dehmelt which provides simple estimates for both the slow and the fast components of the solution. Dehmelt’s approximation also provides a great deal of insight on the behaviour of physical systems described by the Mathieu equation. A crude derivation of the Dehmelt approximation is presented next.

Derivation of Dehmelt's approximation

Comparison of Dehmelt's approximation with the actual solution